All Questions
10,934 questions
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Calculating hyperbolic Fourier series
Question:
is it possible to uniquely express functions locally as infinite sums of hyperbolic sines and cosines
$f(x)=\sum\limits_{i=0}^\infty \alpha_i\sinh(i\cdot x)+\beta_i\cosh(i\cdot x)$
or even ...
- 13.2k
2
votes
0
answers
82
views
What is Lipschitz constant of the radial renormalization $(X,\|\cdot\|_a) \rightarrow (X,\|\cdot\|_b)$ on a normed vector space $X$
Suppose that $X$ is a vector space with two norms $\|\cdot\|_a$ and $\|\cdot\|_b$. The mapping
$$
f(x) = \frac{\|x\|_{a}}{\|x\|_{b}} x, \qquad \forall x \in X,
$$
with $f(0)=0$
is a radial and maps ...
- 5,327
6
votes
1
answer
256
views
Example/Existence of Positive Linear Functional which is NOT Hermitian
We know that if $\mathcal{A}$ is a unital $C^*$-algebra and if $f:\mathcal{A}\to\mathbb{C}$ is a positive linear functional then it is Hermitian. It simply follows from the fact that in $\mathcal{A}$ ...
- 173
0
votes
0
answers
93
views
Orthogonalization of symmetric non-degenerate bilinear forms
It is well-known that given a field $k$ with characteristic different from $2$, every symmetric non-degenerate bilinear form $B$ over a finite-dimensional space can be orthogonalized. This means that ...
- 1,485
2
votes
1
answer
99
views
A question on Bloch functions
Let $\mathcal{B}(\Delta)$ be the space of Bloch functions in the unit disk $\Delta$. For any $f\in \mathcal{B}(\Delta)$, we define the Bloch norm by
$$
\|f\|_{\mathcal{B}}=\sup_{|z|<1}|f'(z)|(1-|z|^...
- 1,706
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votes
0
answers
56
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What is the maximum of $ \frac{\sin(n(x+a))}{\sin(x+a)} + \frac{\sin(n(x-a))}{\sin(x-a)}$?
I have asked this here. Due to inactivity and no satisfying answers, I am asking here. Hope that's okay.
We know the global maxima of the function $\frac{\sin(nx)}{\sin(x)}$
is $n$ (thanks to this ...
- 113
1
vote
1
answer
151
views
Some operators on spheres
Let $S_2$ be the unit sphere in $\mathbb R^3$ equipped with normalized Haar measure. For a continuous function f and $\delta\in (-1,1)$ define $T_\delta f(x):=\int_{\{y:<x,y>=\delta\}}f(y)d_\...
- 1,879
2
votes
1
answer
89
views
The contractivity of the time derivative of the heat semigroup in $L^p$ spaces
Let $M$ be a complete manifold. The heat semigroup $e^{-tL}$ is bounded on $L^p(M)$, for any $1 \leq p \leq \infty$;
see this for instance.
It seems that we can deduce the time derivative of the heat ...
- 121
5
votes
2
answers
258
views
Boundary value of Sobolev space
Let $D$ be a regular domain in $\mathbb R^2$. Suppose that $u \in H_0^1(D) \cap C(D)$. Does this imply $u \in C(\overline D)$ and $u|_{\partial D} = 0$?
- 177
0
votes
0
answers
118
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Find the maximum of an expression under the logconcave assumption
Let $F(v)$ be a cdf over $\left[0,v_{max}\right]$, $1-F(v)$ is logconcave. The corresponding density function is $f(v)$. Let $p^m$ solve $1-F(v)-f(v)v=0$ (it is a FOC of a profit maximization problem)....
1
vote
0
answers
30
views
Generalization of subadditivity analogous to quasiconvexity, and variants
I am curious if there are natural generalizations of subadditivity which have been studied in the past or have been stated in the literature? I (and people that I have talked to) have not had much ...
- 21
2
votes
1
answer
118
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Proving that a polynomial $f(x,y)$ that is unbounded in every direction is bounded below by $1$ outside of a disc of finite radius
This is a follow up from this question.
I have a polynomial function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the ...
- 765
5
votes
2
answers
364
views
Euler-Lagrange equations for minimizer of energy with indicator function
I'm looking for a modern explanation/proof of the derivation of Euler-Lagrange (or first-order or the "first variation") conditions for
$$\min_{u \in H^1_0(\Omega), u \geq 0} \int_\Omega |\...
- 93
1
vote
1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,058
1
vote
1
answer
76
views
Proving that a function $f(x,y)$, that is unbounded in every direction, is uniformly bounded below by $1$ outside some disc of large enough radius
I have a smooth function $f(x,y)$ that is unbounded in every direction. In other words, if we choose a direction $(a,b)\in S^1$ and keep moving along the curve $(ta,tb)$, then $$\lim_{t\to\infty}f(ta,...
- 765
1
vote
1
answer
115
views
Block-diagonal embedding of $U(n)$ into $U(mn)$
What is known about the subgroup $U(n)\subset U(mn)$ for $m,n\in\mathbb{N}$ given by the diagonal embedding
$$ \alpha\mapsto \text{diag}(\alpha,\cdots, \alpha),$$
for $\alpha$ appearing $m$ times?
For ...
2
votes
0
answers
94
views
Nemytskij operator for Lebesgue variable UNBOUNDED exponent spaces
Let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where $\Omega\...
- 1,759
0
votes
1
answer
255
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Carleson's theorem: proof of a lemma
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. At the bottom of page 20 at the beginning of ...
- 75
7
votes
1
answer
252
views
Does a Banach algebra version of "the sum of a closed subspace and a finite dimensional subspace is always closed" exist?
In the setting of Banach spaces, it is well know that if $M$ is a closed subspace of a Banach space $X$ and $F$ is a finite dimensional subspace of $X$, then $M+F$ is closed.
Does a Banach algebra ...
- 515
8
votes
0
answers
177
views
Understanding spaces of negative regularity
I apologize if this question is too basic for this site, but I posted it on mathSE and did not get any responses (link can be found here) so I'm crossposting it here.
Let $C^k(\mathbb{R}^n$) be the ...
- 721
4
votes
1
answer
255
views
Asymptotic behavior and of an integral on a d-dimensional torus
I am trying to evaluate the asymptotic behavior of the following integral as $t \to \infty$:
$$
I(t; \mathbf{v}) = \int_{[-\pi, \pi]^d} \frac{\sin(t f(\mathbf{k}))}{\sin(f(\mathbf{k}))} e^{i t \mathbf{...
- 81
0
votes
0
answers
85
views
Measurable selection for the mean value theorem
When we use the mean value theorem we come across the problem of measurability of the argument. The problem is somehow like that:
Let $f:\Omega\times [0,1]\to\mathbb{R}$ be a Caratheodory function (i....
- 1,759
2
votes
0
answers
84
views
Question about the Nemytsky operator on $L^p$ space
Let $\Omega\subset\mathbb{R}^N$ be a bounded open set, $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function, i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is ...
- 1,759
3
votes
0
answers
147
views
Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
- 91
0
votes
1
answer
123
views
Proving a Fourier transform inequality for functions with mixed variable bounded support
I'm working on a problem involving the Fourier transform and have encountered an inequality that I am unsure how to prove. I would greatly appreciate any help or guidance you can provide.
Let $\gamma\...
- 131
1
vote
0
answers
175
views
Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
- 47
0
votes
0
answers
79
views
Is the Bures metric equivalent to the Euclidean one?
Let $K=\mathbb R$ (reall numbers) or $K=\mathbb C$ (complex numbers). Define $\mathcal M_n$ to be the space of $n\times n$ matrices $A=(a_{i,j})_{1\le i,j\le n}$, with $a_{i,j}\in K$. Let $\|\cdot\|$ ...
- 1,334
1
vote
2
answers
156
views
Numerical evaluation of monomial divided differences
Suppose $f(x)=x^{n+1}$ for some $n\in\mathbb{N}$, and define the divided difference $$f[a,b]=\frac{a^{n+1}-b^{n+1}}{a-b}.$$
I am wondering about the best way to numerically evaluate $f[a,b]$ to high ...
- 45
3
votes
1
answer
227
views
Algebraic and continuous duals of an inverse limit of finite dimensional vector spaces
I have been trying to understand the following section of a paper "Revêtements du demi-plan de Drinfeld et correspondance de Langlands p-adique" by Gabriel Dospinescu and Arthur-César Le ...
- 117
5
votes
1
answer
220
views
How big is the class of all closed range bounded linear operator?
Let $X$ and $Y$ be Banach spaces and let $CR(X,Y)$ denote the set $B(X,Y)$ of all bounded linear maps from $X$ to $Y$ with $T(X)$ closed in $Y$. Certainly $CR(X,Y)$ is not open in $B(X,Y)$ as given ...
- 585
2
votes
1
answer
236
views
A sensible topology on the space of continuous linear maps between Fréchet spaces
Let $V_1$ and $V_2$ be Fréchet spaces. Let $\{ \lVert \cdot \rVert_{1,n} \}_{n \in \mathbb{N}}$ be a family of seminorms for $V_1$ and similarly $\{ \lVert \cdot \rVert_{2,n} \}_{n \in \mathbb{N}}$ ...
- 3,477
2
votes
0
answers
60
views
Semigroup property in SPDEs
In fact, we know that a bounded linear operators on a Banach space $X$ satisfies the semigroup property, i.e. $$S(t+s)=S(t)S(s), \text{for every}\ t,s\geq 0.$$
However, in various literatures, I ...
- 57
1
vote
0
answers
98
views
$(\lambda I-A)^{-1}-(\lambda I-B)^{-1}$ compact implies $\sigma_\text{ess}(A)=\sigma_\text{ess}(B)$
Suppose $H$ is a Hilbert space and $A$, $B$ are two adjoint operators on it (not necessarily bounded), satisfying $D(A)=D(B)$.
Question: If $\exists \lambda\in \rho(A)\cap\rho(B)$ such that $(\lambda ...
- 775
1
vote
1
answer
170
views
Mean of probability distribution
I have a probability distribution defined by the following density function:
$f(k,j,n,m)=\frac{(m n)! \mathcal{S}_k^{(j)}}{(m n)^k (m n-j)!}$ (With $\mathcal{S}_k^{(j)}$ being the Stirling number of ...
- 47
1
vote
1
answer
132
views
Deriving a specific bound for functions in Hardy Space
Reading some article a while ago I read the following: (here $H^2$ represents the Hardy space)
Let $f\in H^2$ be such that $f(0)=1$, and let $0<\lvert\lambda\rvert<1$, then $$\lVert f(\lambda z)\...
1
vote
0
answers
66
views
The derivative of semigroup in the weak sense imply strong sense
Suppose $X$ is a Banach space, and $T(t)$ $t\ge0$ is a strongly continuous semigroup with generator $A$. Assume $\frac{T(t)-I}{t}x$ weakly converges to $y\in X$ when $t\to 0$, then I need to prove $x\...
- 775
7
votes
1
answer
184
views
Functional calculus on the Schwartz space instead of $L^2$?
As far as I know, functional calculus is typically carried out on Hilbert spaces with (possibly unbounded) self-adjoint operators.
However, I wonder if there is a way to do it on the space of test ...
- 3,477
1
vote
0
answers
63
views
Is there any other norms besides cut norm defined on graphon?
Let $\mathcal{W}$ denote the space of all bounded symmetric measurable functions
$W : [0, 1]^2 \rightarrow \mathbb{R}.$ For any $W\in\mathcal{W}$ we say it is a kernel and define its cut norm $\lVert ...
- 349
5
votes
2
answers
432
views
Does closedness of the image of unit sphere imply the closed range of the operator
Let $X$ and $Y$ be Banach spaces and let $T:X\to Y$ be a bounded linear operator such that $T(S_X)$ is closed in $Y$. Does it imply that $T(X)$ is closed? Any hint is appreciated.
- 585
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votes
0
answers
66
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convolution of the fundamental solution with the homogeneous solution
I have a question about the convolution of the fundamental solution with the homogeneous solution. Namely if the 2 are convoluble then the homogeneous solution is necessarily zero?
Let $U$ and $E$ ...
-1
votes
2
answers
250
views
$p$-norm of random variables and weighted $L^p$ space resemblance
I noticed a very similar relationship between weighted $L^p$ space (denoted $L_w^p$) and normed vector space of random variables. I want to unify these two spaces but there always seems to be a ...
- 1
7
votes
0
answers
131
views
Approximation of a continuous curve on commuting matrices
I have a continuous curve $A:\mathbf{R}_+\rightarrow \text{M}_N(\mathbf{R})$ such that
$[A(t),A(s)] \operatorname*{\longrightarrow}_{t,s\rightarrow +\infty} 0$, where $[A(t),A(s)] = A(t)A(s)-A(s)A(t)$....
- 3,425
2
votes
0
answers
96
views
Isometric Schröder-Bernstein theorem for injective Banach spaces?
It's known that every injective Banach space is of the form $C(M)$ where $M$ is a compact, Hausdorff, extremally disconnected topological space.
Let $X$, $Y$ be two injective Banach spaces such that,
...
- 2,605
2
votes
0
answers
102
views
Existence of unique-up-to-shift solution of a Volterra equation
Let $\Delta=\{(t,s):\ 0<s\leq t\leq1\}$, and suppose $k:\Delta\to\mathbb R$ and $f:(0,1]\to\mathbb R$ are continuous. Further assume that for every $t\in(0,1]$, the function $k(t,\cdot):(0,t]\to\...
- 537
2
votes
0
answers
71
views
How to naturally define an output space with certain properties
Consider the following regression problem $v=A(u) + \varepsilon$
for some operator $A:\mathcal{U} \rightarrow \mathcal{V}$ and some function spaces $\mathcal{U},\mathcal{V}$, mapping from $\mathcal{X}$...
- 51
7
votes
2
answers
350
views
Can the Banach algebra structure on $B(E)$ be (almost) retrieved from its Banach space structure?
This is basically just out of curiosity. Also, since my research area is in von Neumann algebras and my knowledge of general Banach algebras as well as general Banach spaces is somewhat limited, I ...
- 2,830
1
vote
1
answer
87
views
Convergence in $H^{-2}$ of $L^2$-functions with limit in $L^2$
Assume a sequence $f_n$ in $L^2(\mathbb{R}^d)$ converges in $H^{-2}$ (w.r.t. its norm topology) to a limit $f \in L^2(\mathbb{R}^d)$. In this case, can one improve the convergence, for instance to ...
- 133
5
votes
2
answers
517
views
Functions whose product with every $L^1$ function is $L^1$
Let $\mu$ be a probability measure and $f$ a measurable function whose
product with any integrable function is integrable: $$
\int|g|\,{\rm{d}}\mu<\infty\implies \int|fg|\,{\rm{d}}\mu<\infty. $$
...
- 3,599
2
votes
0
answers
66
views
interchange of integrals and semigroup without the semigroup being an integral operator
In Cazenave's book: BREZIS, HAIM.; CAZENAVE, T. Nonlinear evolution equations. IM-UFRJ, Rio, v. 1, p. 994, 1994. The following corollary appears
The formula (1.5.2) is Duhamel formula:
$$u(t) = T(t)u(...
- 677
2
votes
1
answer
211
views
Hölder continuity in time of heat semigroup for regular initial distribution
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
Let $(p_t)_{t>0}$ be the standard Gaussian heat kernel on $\bR^d$, i.e.,
$$
p_t (x) := \frac{1}{(4 \pi t)^{\frac{d}{2}}} \...
- 835